This invention relates to a radially wedging, frictional overrunning clutch having z sprags of a profiled section (z being the number of sprags) which are arranged in a circumferential series between a cylindrical inner track having a diameter D.sub.i =2R.sub.i and a cylindrical outer track which is arranged concentrically with respect to the inner cylindrical track and which has a diameter D.sub.a =2R.sub.a. The inner cylindrical track is formed by the outer cylinder face of an inner clutch ring whereas the outer track is formed by the inner cylinder face of an outer clutch ring. The sprags are disposed in such a manner that upon rotation of the clutch rings relative to one another in the locking direction, the sprags wedge with their clutch faces against the cylindrical tracks and generate, at the line of contact between sprag and inner track, a radially outwardly directed normal force N.sub.i in the inner ring and further generate at the line of contact between sprag and outer track, a radially inwardly directed normal force N.sub.a in the outer ring. The clutch faces of the sprags have at the line of contact with the inner or, as the case may be, the outer track, a radius of curvature r.sub.i and r.sub.a, respectively. The distance between the centers of the two curvatures is designated at c. The inner wedging angle .epsilon..sub.i and the outer wedging angle .epsilon..sub.a between the plane containing the two lines of contact and the plane containing one of these lines of contact and the rotary axis of the overrunning clutch are determined by the following equations: ##EQU1## AND, RESPECTIVELY, ##EQU2##
In this manner, the torques T.sub.i and T.sub.a related to the inner ring and the outer ring, respectively, are obtained as EQU T.sub.i = z N.sub.i R.sub.i tan .epsilon..sub.i (3)
and, respectively, EQU T.sub.a = z N.sub.a R.sub.a tan .epsilon..sub.a, (4)
wherein EQU T.sub.i = T.sub.a = T.
The relative rotation between the inner ring and the outer ring, occurring during application of load, is designated by rotational angle .alpha..
The computation of sprag-type overrunning clutches is effected in a conventional manner with the aid of the above-given or related relationships. For a more detailed explanation of these relationships reference is made to FIG. 1 which shows a fragmentary radial section of an overrunning clutch with the more important forces appearing upon torque transmission at the sprag. The outer cylindrical surface of the inner ring 1 constitutes the inner sprag track 2, while the inner cylindrical surface of the outer ring 3 constitutes the outer sprag track 4. Between the inner ring and the outer ring there are positioned the circumferentially arranged sprags 5 which can wedge with their inner clutch face 6 against the inner track 2 and with their outer clutch face 7 against the outer track 4. Upon such an occurrence the forces illustrated in FIG. 1 are generated. In the lines of contact between the sprags and the sprag tracks there act the above-mentioned normal forces N.sub.i and N.sub.a and the circumferential forces H.sub.i and H.sub.a. In order to ensure an equilibrium of force, the resultants of N.sub.i and H.sub.i and, respectively, N.sub.a and H.sub.a have to lie on the same line of action, must be oppositely oriented and must be of identical magnitude, as illustrated in FIG. 1. If one considers that EQU H.sub.i = N.sub.i tan.epsilon..sub.i
and, respectively, EQU H.sub.a = N.sub.a tan.epsilon..sub.a,
for the torque T = T.sub.i = T.sub.a there can be obtained immediately the relationships (3) and (4) set forth earlier. It is noted that .epsilon..sub.i and .epsilon..sub.a are structurally predetermined magnitudes which may be constant or may have, in the wedging zone, a minimum value as disclosed in German Laid-Open Application (Offenlegungsschrift) No. 2,204,305 and German Pat. No. 1,199,066. The computation of .epsilon..sub.i and .epsilon..sub.a may be effected trigonometrically with the aid of equations (1) and (2), respectively.
When the overrunning clutch is placed under load in the coupling direction, there is effected a relative rotation between the inner ring and the outer ring of the overrunning clutch. During this relative rotation, the magnitude of which is dependent from the torque applied to the overrunning clutch, the sprags wedge to a greater or lesser extent against the spring tracks of the rings and in this manner transmit the torque from one ring to the other. If in a coordinate system one plots the torque T versus the relative rotation between the inner and outer ring, designated as .alpha., a curve is obtained which, similar to the rotationally elastic clutches, is designated as a torsion spring curve because it indicates the relationship between the load torque and the angle of rotation.
In overrunning clutches which are used as indexing units, the torsion spring curve has a significant effect on the accuracy of indexing. In practice, the driving torque oscillates with a certain deviation about a desired value because of varying properties (thickness, strength, coefficient of friction, radius of curvature, etc.) of the material to be fed. Accordingly, the angle of rotation between the driving and driven component of the overrunning clutch also changes, resulting in an irregular material feed. In many cases, however, it is important that the advance rate be constant, that is, the torsion spring curve be as steep as possible, since the steeper this spring curve, the lesser the variation of the angle of rotation.
In the design of sprag clutches one has generally limited oneself to ensure, with the aid of the above-given relationships, that the values of the wedging angle are always below the maximum permissible coefficient of friction and that Hert's stress between the sprag and the track does not exceed the maximum permissible value. With regard to the generation of the spring curves of predetermined slope, heretofore only empirical values were available. The reason for this is that the rotation of the overrunning clutch rings with respect to one another is based on the elastic deformations of the clutch components and such deformations could heretofore not be determined mathematically. A known method of computation for the torsion spring curve applies only to overrunning clutch rings having a very small wall thickness. Such an initial condition, however, is not present in overrunning clutches operating with sprags. Further, the variation of the wedging angle as a function of increasing angle of rotation is not taken into account. Such known method is discussed in the work by C. B. Biezeno and R. Grammel, entitled TECHNISCHE DYNAMIK, volume 1, 2nd edition (publisher: Springer, 1953).